Proximal causal inference for conditional separable effects

Scientists regularly pose questions about treatment effects on outcomes conditional on a posttreatment event. However, causal inference in such settings requires care, even in perfectly executed randomized experiments. Recently, the conditional separable effect (CSE) was proposed as an interventionist estimand that corresponds to scientifically meaningful questions in these settings. However, existing results for the CSE require no unmeasured confounding between the outcome and posttreatment event, an assumption frequently violated in practice. In this work, we address this concern by developing new identification and estimation results for the CSE that allow for unmeasured confounding. We establish nonparametric identification of the CSE in observational and experimental settings with time-varying confounders, provided that certain proxy variables for hidden common causes of the posttreatment event and outcome are available. For inference, we characterize an influence function for the CSE under a semiparametric model where nuisance functions are a priori unrestricted. Using modern machine learning methods, we construct nonparametric nuisance function estimators and establish convergence rates that improve upon existing results. Moreover, we develop a consistent, asymptotically linear, and locally semiparametric efficient estimator of the CSE. We illustrate our framework with simulation studies and a real-world cancer therapy trial.